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Figure 1:
The inverse of migration rates vs.
eccentricity for four planet masses. Top left panel - 1 ![]() ![]() ![]() ![]() |
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Figure 2:
Eccentricity damping rates
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Figure 3:
Migration time (computed from Eq. (11))
vs. softening for a body of ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Figure 4: Evolution of a 10 planet N-body model with the fiducial setup (see main text). Top: semi-major axes of the migrating embryos. A brief period of activity is followed by a long period of migration between bodies in first-order mean-motion resonances. Bottom: the embryos' eccentricities over the same time. Damping from the disc prevents eccentricities from growing outside periods of intense activity. |
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Figure 5:
First order resonances between five of the bodies in
Fig. 4. Each plot is labelled (P1, P2)-p:q, where P1/P2 are
the exterior/interior protoplanets respectively, and p:q gives the
commensurability between them; left/right columns show the resonant argument
associated with the exterior/interior body, respectively. Each plot takes the
same colour as the corresponding exterior protoplanet in that resonance.
Note that the y-axis differs by ![]() |
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Figure 6:
As Fig. 4, but for a model with a higher initial
eccentricity distribution (
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Figure 7:
As Fig. 4, but with initial separations
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Figure 8:
As Fig. 4, but with a Gaussian mass distribution for
the protoplanets. From the object at smallest initial radius, to the
nearest 0.1 ![]() ![]() ![]() ![]() |
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Figure 9:
As Fig. 4, but for a model with 20 protoplanets
separated initially by 5
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Figure 10: As Fig. 4, but for a model with reduced (gaseous) disc mass. Eccentricity damping (and migration rates) are reduced by a factor of 5, but collisions reduce the total number of bodies while eccentricity damping is still too strong for mutual interactions to excite the population. |
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Figure 11:
As Fig. 4, but for a model with eccentricity damping
(Eq. (17)) reduced by a factor of 50. Scattering of only ten bodies is now
self-sustaining over
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Figure 12: Evolution of a model of ten bodies using the NIRVANA hydrodynamic code, similar to Fig. 4. Collisions remove smaller, easily perturbed bodies as differential migration produces a successively more crowded system, until stable resonances form and the group migrate inwards at a single rate. As in the majority of N-body integrations, no body achieves and sustains ep>0.1. |
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Figure 13:
As Fig. 12, but with higher initial eccentricities,
similar to Fig. 6. All surviving bodies end the run in 5:4 and/or 6:5
resonances with adjacent protoplanets, while a co-orbital system forms and
remains stable until the end of integrations, over
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Figure 14: First order resonances between the protoplanets in Fig. 13. Each plot is labelled and coloured as in Fig. 5. Note that the trojan planets (P2=2,4) both share the two resonances with exterior bodies (P1=5,6), as shown in the lower four pairs of plots. |
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Figure 15:
As Fig. 12, but with reduced initial separations
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Figure 16:
As Fig. 15, but with the increased initial
eccentricities of Fig. 6 (
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Figure 17:
As Fig. 15, but with with a randomised, Gaussian protoplanet mass distribution, similar to Fig. 8. Working radially outwards through the initial distribution, to the nearest 0.1 ![]() ![]() ![]() |
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Figure 18: A series of surface density plots of the simulation shown in Fig. 12. The time in years of each snapshot is shown above each plot. The protoplanets are represented by white circles, with size proportional to mass. |
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